Let $T = t(0) t(1) ... t(n) ... \in \{0,1\}^\mathbb{N}$ be the Thue-Morse sequence (i.e. the infinite word $T$ obtained as the limit in $\{0,1\}^\mathbb{N}$ of the sequence of finite words $(T_r)_{r \in \mathbb{N}}$ defined by the recursion $T_0 = 0 , T'_0 = 1$ and $T_{r+1} = T_r T'_r , T'_{r+1} = T'_r T_r$ for any non negative integer $r$).

The Thue-Morse sequence is a well known example of an almost periodic and zero entropy deterministic binary sequence.

The goal of this talk is to show that its subsequence along square numbers $(t(n^2))_{n \in \mathbb{N}}$ is normal, which mean in some sense quasi-random.

(joint work with Michael Drmota and Joel Rivat).